Rock physics modeling plays a critical role in seismic inversion and interpretation by providing a critical link between petrophysical properties (e.g., porosity, shale volume, water saturation, etc.) and seismic properties (e.g., P- and S-wave velocities, attenuation and frequency content). It is, therefore, extremely important to build an accurate and robust rock physics model that represents the combined effect of the key controlling factors, such as porosity, pore geometry, pore connectivity, fluid type (or water saturation), clay content, mineralogy, stress, temperature and anisotropy.
It is well known that sedimentary rocks (the type of rock where petroleum can often be found) are anisotropic in nature. That is, their elastic properties vary with direction. Recent published literature shows that seismic anisotropy is a primary factor controlling the amplitude-versus-offset (AVO) behavior. In a CMP (common-middle point) gather, the amplitude of a seismic reflection from a particular interface varies with offset or incident angle. The behavior of the amplitude change with offset can be linked to fluid content in the rock. For example, typical gas sands overlain by shale are often characterized by a “Class III” type AVO, i.e., negative polarity at the interface and increasing amplitude with offset. Oil sands may display Class II type AVO, i.e., very weak amplitude at the near offset and strong amplitude at the far offset. The present inventors have found that the anisotropy effect is particularly important for high-impedance sands (the acoustic impedance of the sands is higher than the acoustic impedance of their surrounding shale). FIG. 1 shows P-wave reflectivity over a single interface as a function of incident angle using both isotropic 11 and anisotropic 12 earth models. It clearly demonstrates that anisotropy is an important factor for AVO modeling. This is particularly true for incident angles larger than 30 degrees. The results also demonstrate that a theoretical rock physics model with anisotropic capabilities is highly desirable.
There are two important types of anisotropy in clastic sedimentary rocks: (1) shale anisotropy and (2) stress-induced anisotropy. Shale anisotropy is common in clastic rocks. Shale anisotropy is the most common anisotropy in sedimentary rock. This is simply because over 70% of sedimentary rocks are shale. Shale anisotropy is caused by the preferred orientation of the pore space between the clay particles. Stress-induced anisotropy is caused by the differential stresses in the earth crust. In general, the three principal stresses are often different from each other due to the tectonic movement of the earth crust and overburden effect. In a typical relaxed sedimentary basin, assuming negligible tectonic movement, the vertical stress is often much higher than the two horizontal stresses. In the major principal stress direction, the rock is compressed more in comparison with the compression at the other two directions. This differential compaction will result in a differential closure of soft pore, or cracks, in the rock. Cracks aligned perpendicular to the major principal stress have a higher tendency of being closed than cracks aligned in other directions. Consequently, the compressional wave will travel faster in the major principal stress direction.
In general, shale anisotropy is strong (higher than 10%). Seismic anisotropy is typically measured using Thomsen parameters, ε, γ and δ, which are defined as follows:
      ɛ    =                            C          11                -                  C          33                            2        ⁢                                  ⁢                  C          33                          γ    =                            C          66                -                  C          44                            2        ⁢                                  ⁢                  C          44                          δ    =                                        (                                          C                13                            +                              C                44                                      )                    2                -                              (                                          C                33                            -                              C                44                                      )                    2                            2        ⁢                                  ⁢                              C            33                    ⁡                      (                                          C                33                            -                              C                44                                      )                              Here, Cij is the elastic tensor of the anisotropic rock. The quantity ε approximately measures the P-wave anisotropy, i.e. the relative change between P-wave velocity in the fast direction and that in slow direction. Similarly, γ measures shear-wave anisotropy. δ controls the P-wave and SV-wave velocity profiles at intermediate angles between the fast and slow direction. Typically, ε or δ is used to quantify seismic anisotropy. A characterization of 20% anisotropy thus means that the P- or S-wave velocity in the fast direction is about 20% faster than that in the slow direction.
Because shale anisotropy is strong, it has a larger effect than stress-induced anisotropy on AVO modeling. Stress-induced anisotropy is commonly seen in shallow unconsolidated sands where the vertical effective stress can be significantly higher than the horizontal effective stresses. In cases where there is little tectonic movement, stress-induced anisotropy cannot be measured using cross-dipole logs since the two horizontal stresses are more or less the same. In areas where the tectonic movement is large, azimuthal anisotropy exists and can be measured using cross-dipole logs. Unlike shale anisotropy, stress-induced anisotropy is more difficult to predict because the principal stresses are controlled by many factors including overburden, tectonic, local structures, fault systems and rock properties.
Various theories called effective medium theories have been proposed to simulate shale or stress-induced anisotropy in rocks. Some of these are briefly discussed below. Most of these theories ignore the mechanical interaction between the pores/cracks. These effective medium theories are therefore valid only for dilute concentration of pores. This limitation makes the first-order theories of little practical use. The problem can be resolved using the differential effective medium (DEM) scheme or the self-consistent (SC) scheme. DEM is discussed by Nishizawa in Journal of Physical Earth 30, 331-347 (1982) and by Hornby, et al. in Geophysics 59, 1570-1583 (1994). SC is discussed by Hill in Journal of Mechanics and Physics of Solids 13, 213-222 (1965) and by Willis in J. Mech. Phys. Solids 25, 185-202. (1977). However, either scheme drastically slows down the numerical computation. It would be desirable, to speed up the calculations while maintaining the accuracy of the method. Furthermore, there is no model that can handle both shale anisotropy and stress-induced anisotropy simultaneously. The present invention satisfies both of these needs.
Empirical rock physics models are widely used in the industry due to their simplicity. These empirical models typically assume a linear relationship between P-wave (or S-wave) velocity, porosity and/or shale volume. Despite some limited success of such models, there are increasing concerns about their applicability to seismic inversion (i.e., solving for petrophysical properties using seismic data) since they are data-driven. Without a large amount of data to calibrate these empirical models, they often provide incorrect, sometimes even misleading, results. In many exploration and/or development circumstances, one often does not have the data needed for the calibration. Another major drawback with the empirical models is that they provide little physical insight. For example, one may find a simple relationship between permeability and velocity and conclude that permeability is a major controlling factor for velocity when, in fact, the velocity change is largely caused by porosity. A good correlation between porosity and permeability makes permeability look like a controlling factor for velocity. Permeability matters, but usually has a secondary effect on velocity. Finally, an empirical model can only handle a very limited number of factors, typically fewer than three. For comprehensive rock physics modeling, one needs to consider the combined effect of porosity, pore type, shale volume, fluid content (water saturation), fluid communication, pressure, temperature and frequency.
There are a limited number of theoretical rock physics models in the literature. For example, M. A. Biot, “Theory of propagation of elastic waves in a fluid saturated porous solid,” Journal of Acoustic Society of America 28, 168-191 (1956); G. T. Kuster and M. N. Toksoz, “Velocity and attenuation of seismic waves in two-phase media, Part 1: Theoretical formulation,” Geophysics 39, 587-606 (1974). In general, these models can be used to explain the elastic behavior observed in the laboratory. However, it is often difficult to apply these models to real cases (e.g., well logs). Xu and White developed a practical model that simulates the combined effect of a number of factors on P- and S-wave velocities. (“A new velocity model for clay-sand mixtures,” Geophysical Prospecting 43, 91-118 (1995); and “A physical model for shear-wave velocity prediction,” Geophysical Prospecting 44, 687-717 (1996)) But the model does not handle the anisotropy effect. Hornby, et al. propose an effective medium model to simulate shale anisotropy (Geophysics 59, 1570-1583 (1994)). But, their model is limited to pure shales only, not applicable to sandy shales or shaly sands. This greatly limits the applicability of their model since sedimentary rocks are made of not only shales but also other rock types, such as sandstone, siltstone, limestone, etc. Also, Hornby's approach is valid for high frequencies only and it is, therefore, always Gassmann-inconsistent. Gassmann's theory has been widely used in the oil industry for fluid substitution (hypothetically substituting pore fluid in reservoir rocks from one type to another, e.g., from water to oil). One key assumption in Gassmann's theory is that the frequency of the seismic wave is low enough so that pore pressure has ample time to be equilibrated. Therefore, any model which gives the low frequency response or equalized pore pressure is called Gassmann-consistent.
Keys and Xu (Geophysics 67, 1406-1414 (2002)) propose a dry rock approximation method, which dramatically speeds up the numerical calculation of the differential effective medium scheme while maintaining its accuracy. Unfortunately, the proposed method does not work for the anisotropic case.
Fluid substitution is an important topic for seismic identification of reservoir fluid. Traditionally, different fluid phases, e.g. gas and brine, are mixed using the Wood Suspension law before they are put into the rock using Gassmann (1951) equations. This approach puts fluid mixtures uniformly into all the pore space, regardless of the pore size, wettability and permeability of the rock. Laboratory measurements demonstrate that the approach is probably applicable to rocks with relatively high permeability and at relatively high effective stresses where micro-cracks are closed. It is highly questionable if the approach is valid in shaly sands, in which micro-pores tend to be water-wet due to the capillary effect.
Thus, there is a need for developing an anisotropic rock physics model, which has a fundamental physical basis. The model should be Gassmann-consistent and correctly treat the capillary effect on different fluid phase distribution at pore scale. The model should be accurate and efficient enough to be applied to well log analysis and/or seismic inversion. In particular, the model should be able to handle different kinds of anisotropy (e.g., shale and stress-induced anisotropy). The present invention fills this need.